1.5.9 Graph Isomorphism

INPUT OUTPUT

Problem:
Find a (all) mappings f of the vertices of g to the vertices
of h such that g and h are identical, ie. (x,y) is an
edge of g iff (f(x),f(y)) is an edge of h.

Excerpt from
The Algorithm Design Manual:
Isomorphism is the problem of testing whether two graphs are really the
same.
Suppose we are given a collection of graphs and must perform some operation
on each of them.
If we can identify which of the graphs are duplicates,
they can be discarded so as to avoid redundant work.

We need some terminology
to settle what is meant when we say two graphs are the same.
Two labeled graphs G=(V_g,E_g) and H=(V_h,E_h)
are identical when (x,y) \in Eg iff (x,y) \in E_h.
The isomorphism problem consists of finding a mapping from the vertices
of G to H such
that they are identical.
Such a mapping is called an isomorphism.

Identifying symmetries is another important application of graph isomorphism.
A mapping of a graph to itself is called an automorphism, and the
collection of automorphisms (its automorphism group)
provides a great deal of information about symmetries in the graph.
For example, the complete graph K_n has n! automorphisms (any mapping will
do), while an arbitrary random graph is likely to have few or perhaps
only one, since G is always identical to itself.